Since \(\dim (A^{\prime })\ge \alpha -\varepsilon /20\)  and \(\dim (B^{\prime })\ge \alpha -\varepsilon /20\) , by Frostman's lemma (see e.g. [1]}), there are Borel measures \(\mu \)  and \(\nu \) , supported on \(A^{\prime }\)  and \(B^{\prime }\)  respectively, with \(\mu (A^{\prime })>0,\)  and \(\nu (B^{\prime })>0\) , so that
\(\begin{split}&\mu (A\cap I)\le |I|^{\alpha -\varepsilon /10}\ \textrm {for every interval}\ I,\\&\nu (B\cap J)\le |J|^{\alpha -\varepsilon /10}\ \textrm {for every interval}\ J.\end{split}\) 
